The confluent hypergeometric differential equation
has a regular singular point at
and an essential singularity at
. Solutions analytic at
are confluent hypergeometric functions of the first kind (or Kummer functions):
,
where
are Pochhammer symbols defined by
,
,
. For
, the function becomes singular, unless
is an equal or smaller negative integer (
), and it is convenient to define the regularized confluent hypergeometric
, which is an entire function for all values of
,
and
.
The second, linearly independent solutions of the differential equation are confluent hypergeometric functions of the second kind (or Tricomi functions), defined by
, where the generalized hypergeometric function
represents a formal asymptotic series.
If the hypergeometric function with argument
is complex, both the real and imaginary parts are plotted (black and red curves).
For certain choices of the parameters
and
, the hypergeometric functions are related to various transcendental and special functions. Several illustrations are given in the snapshots.
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